[PPT] - Finite Element Multigrid Framework for Mimetic Finite Difference PowerPoint Presentation

SLIDE 1

Finite Element Multigrid Framework for Mimetic Finite Difference Discretizations

Xiaozhe Hu

Tufts University

Polytopal Element Methods in Mathematics and Engineering, October 26 - 28, 2015

Joint work with: F.J. Gaspar, C. Rodrigo (Universidad de Zaragoza), and L. Zikatanov (Penn State)

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

1 / 25

SLIDE 2

Outline

1 Introduction

2 Relation Between Finite Element and Mimetic Finite Difference

3 Geometric Multigrid Methods

4 Conclusions and Future Work

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

2 / 25

SLIDE 3

Introduction

Outline

1 Introduction

2 Relation Between Finite Element and Mimetic Finite Difference

3 Geometric Multigrid Methods

4 Conclusions and Future Work

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

3 / 25

SLIDE 4

Introduction

Model Problems:

Model Equations curl rotu + κu = f, in Ω −grad divu + κu = f, in Ω

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

4 / 25

SLIDE 5

Introduction

Model Problems:

Model Equations curl rotu + κu = f, in Ω −grad divu + κu = f, in Ω

Applications: Darcy’s flow, Maxwell’s equation, etc.
Involve special physical and mathematical properties: mass conservation,

Gauss’s Law, exact sequence property of the differential operators, etc.

Complicated geometry: unstructured triangulation, polytopal mesh, etc.
X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

4 / 25

SLIDE 6

Introduction

Model Problems:

Model Equations curl rotu + κu = f, in Ω −grad divu + κu = f, in Ω

Applications: Darcy’s flow, Maxwell’s equation, etc.
Involve special physical and mathematical properties: mass conservation,

Gauss’s Law, exact sequence property of the differential operators, etc.

Complicated geometry: unstructured triangulation, polytopal mesh, etc.

Structure-preserving discretizations on polytopal meshes are preferred!!

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

4 / 25

SLIDE 7

Introduction

Model Problems:

Model Equations curl rotu + κu = f, in Ω −grad divu + κu = f, in Ω

Applications: Darcy’s flow, Maxwell’s equation, etc.
Involve special physical and mathematical properties: mass conservation,

Gauss’s Law, exact sequence property of the differential operators, etc.

Complicated geometry: unstructured triangulation, polytopal mesh, etc.

Structure-preserving discretizations on polytopal meshes are preferred!!

Mimetic finite difference method (Lipnikov, Manzini, & Shashkov 2014; Beir˜

ao Da Veiga, Lipnikov, & Manzini 2014;...)

Generalized finite difference method (Bossavit 2001; 2005; Gillette & Bajaj 2011; ...)
Mixed finite element method (Brezzi & Fotin 1991; ...)
Finite element exterior calculus (Arnold, Falk, & Winther 2006; 2010; ...)
Discontinuous Galerkin method (Arnold, Brezzi, Cockburn, & Marini 2002; ...)
Virtual element method (Beir˜

ao Da Veiga, Brezzi, Cangiani, Manzini, Marini & Russo 2013; ...)

Weak Galerkin method (Wang & Ye 2013; ...)
Hybrid High-Order method (Di Pietro, Ern, & Lemaire 2014; ...)
...
X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

4 / 25

SLIDE 8

Introduction

Motivation

A question: How to solve Ax = b efficiently

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

5 / 25

SLIDE 9

Introduction

Motivation

A question: How to solve Ax = b efficiently This talk:

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

5 / 25

SLIDE 10

Introduction

Motivation

A question: How to solve Ax = b efficiently This talk:

focus on mimetic FDM (Vector Analysis Grid Operators Method, Vabishchevich, 2005)
X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

5 / 25

SLIDE 11

Introduction

Motivation

A question: How to solve Ax = b efficiently This talk:

focus on mimetic FDM (Vector Analysis Grid Operators Method, Vabishchevich, 2005)
show relation between mimetic FDM and FEM
X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

5 / 25

SLIDE 12

Introduction

Motivation

A question: How to solve Ax = b efficiently This talk:

focus on mimetic FDM (Vector Analysis Grid Operators Method, Vabishchevich, 2005)
show relation between mimetic FDM and FEM
design geometric multigrid methods for mimetic FDM
X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

5 / 25

SLIDE 13

Introduction

Motivation

A question: How to solve Ax = b efficiently This talk:

focus on mimetic FDM (Vector Analysis Grid Operators Method, Vabishchevich, 2005)
show relation between mimetic FDM and FEM
design geometric multigrid methods for mimetic FDM

Relation between MFD and MFEM for diffusion (Berndt, Lipnikov, Moulton, & Shashkov 2001;

Berndt, Lipnikov, Shashkov, Wheeler & Yotov 2005; Droniou, Eymard, Gallou¨ et, & Herbin 2010)

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

5 / 25

SLIDE 14

Introduction

Mimetic FDM: Delaunay and Voronoi Grids

Computational domain: Ω = Ω ∪ ∂Ω Acute Delaunay grid {xD

i , i = 1, . . . , ND}

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

6 / 25

SLIDE 15

Introduction

Mimetic FDM: Delaunay and Voronoi Grids

Computational domain: Ω = Ω ∪ ∂Ω Acute Delaunay grid {xD

i , i = 1, . . . , ND}

Dual mesh: Voronoi grid Voronoi points: centers of the circumscribed circles on each triangle {xV

k , i = 1, . . . , NV }

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

6 / 25

SLIDE 16

Introduction

Mimetic FDM: Delaunay and Voronoi Grids

Computational domain: Ω = Ω ∪ ∂Ω

D
D
Acute Delaunay grid

{xD

i , i = 1, . . . , ND}

Dual mesh: Voronoi grid Voronoi points: centers of the circumscribed circles on each triangle {xV

k , i = 1, . . . , NV }

For each xD

i

Voronoi polygon: Vi = {x ∈ Ω | |x − xD

i | < |x − xD j |, j = 1, . . . , ND, j = i},

and we denote: ∂Vij = ∂Vi ∩ ∂Vj

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

6 / 25

SLIDE 17

Introduction

Mimetic FDM: Grid Functions

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

7 / 25

SLIDE 18

Introduction

Mimetic FDM: Grid Functions

Scalar Grid Functions:
Delaunay grid: u(x) are defined by u(xD

i ) = uD i

at the nodes xD

i . HD denotes the set of u(x).

Voronoi grid: u(x) are defined by u(xV

k ) = uV k at the nodes

xV

k . HV denotes the set of u(x).

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

7 / 25

SLIDE 19

Introduction

Mimetic FDM: Grid Functions

Scalar Grid Functions:
Delaunay grid: u(x) are defined by u(xD

i ) = uD i

at the nodes xD

i . HD denotes the set of u(x).

Voronoi grid: u(x) are defined by u(xV

k ) = uV k at the nodes

xV

k . HV denotes the set of u(x).

Vector Grid Functions:
Delaunay grid: u(x) are defined by u(x) · eD

ij = uD ij at the

middle point of the edges xD

ij = 1 2(xD i + xD j ). HD denotes the

set of u(x)

Voronoi grid: u(x) are defined by u(x) · eV

km = uV km at the

intersect points. HV denotes the set of u(x)

eD

ij is directed from the node with smaller index to the node with larger index

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

7 / 25

SLIDE 20

Introduction

Mimetic FDM: Discrete Operators

D
D
D
V
D
D
D
D
D
V
V
V
D
D
X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

8 / 25

SLIDE 21

Introduction

Mimetic FDM: Discrete Operators

Discrete Gradient Operators: gradh : HD → HD (gradh u)D

ij :=η(i, j)uD j − uD i

lD

ij

, with η(i, j) = 1, if j > i −1, if j < i

D
D
D
V
D
D
D
D
D
V
V
V
D
D
X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

8 / 25

SLIDE 22

Introduction

Mimetic FDM: Discrete Operators

Discrete Gradient Operators: gradh : HD → HD (gradh u)D

ij :=η(i, j)uD j − uD i

lD

ij

, with η(i, j) = 1, if j > i −1, if j < i Discrete Rotor Operator: roth : HD → HV (roth u)V

k = η(i, j) uD ij lD ij + η(j, l) uD jl lD jl + η(l, i) uD li lD li

meas(Dk)

D
D
D
V
D
D
D
D
D
V
V
V
D
D
X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

8 / 25

SLIDE 23

Introduction

Mimetic FDM: Discrete Operators

Discrete Gradient Operators: gradh : HD → HD (gradh u)D

ij :=η(i, j)uD j − uD i

lD

ij

, with η(i, j) = 1, if j > i −1, if j < i Discrete Rotor Operator: roth : HD → HV (roth u)V

k = η(i, j) uD ij lD ij + η(j, l) uD jl lD jl + η(l, i) uD li lD li

meas(Dk) Discrete Curl Operator: curlh : HV → HD (curlh u)D

ij = η(k, m)uV k − uV m

lV

km

D
D
D
V
D
D
D
D
D
V
V
V
D
D
X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

8 / 25

SLIDE 24

Introduction

Mimetic FDM: Discrete Operators

Discrete Gradient Operators: gradh : HD → HD (gradh u)D

ij :=η(i, j)uD j − uD i

lD

ij

, with η(i, j) = 1, if j > i −1, if j < i Discrete Rotor Operator: roth : HD → HV (roth u)V

k = η(i, j) uD ij lD ij + η(j, l) uD jl lD jl + η(l, i) uD li lD li

meas(Dk) Discrete Curl Operator: curlh : HV → HD (curlh u)D

ij = η(k, m)uV k − uV m

lV

km

Discrete Divergence Operator: divh : HD → HD (divh u)D

i =

1 meas(Vi)

j∈WV (i)

uD

ij (eD ij · nV ij ) meas(∂Vij)

D
D
D
V
D
D
D
D
D
V
V
V
D
D
X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

8 / 25

SLIDE 25

Introduction

Mimetic FDM: Stencils

1/ 2/3 2/3 2/3 2/3 2/3 2/3 1/ 4√3/3 4√3/3 4√3/3 √3/ √3/

gradh divh roth curlh

8/ 4/ 4/ 4/ 4/ 4/3 2/3 2/3 2/3 2/3 2/3 2/3 2/3 2/3 2/3 2/3

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

9 / 25

SLIDE 26

Introduction

Mimetic FDM: Stencils

1/ 2/3 2/3 2/3 2/3 2/3 2/3 1/ 4√3/3 4√3/3 4√3/3 √3/ √3/

gradh divh roth curlh Mimetic FDM curlh rothuh + κuh = fh, in Ω −gradh divhuh + κuh = fh, in Ω

8/ 4/ 4/ 4/ 4/ 4/3 2/3 2/3 2/3 2/3 2/3 2/3 2/3 2/3 2/3 2/3

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

9 / 25

SLIDE 27

Introduction

Mimetic FDM: Stencils

1/ 2/3 2/3 2/3 2/3 2/3 2/3 1/ 4√3/3 4√3/3 4√3/3 √3/ √3/

gradh divh roth curlh Mimetic FDM curlh rothuh + κuh = fh, in Ω −gradh divhuh + κuh = fh, in Ω

8/ 4/ 4/ 4/ 4/ 4/3 2/3 2/3 2/3 2/3 2/3 2/3 2/3 2/3 2/3 2/3

curlh roth − gradh divh

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

9 / 25

SLIDE 28

Relation Between Finite Element and Mimetic Finite Difference

Outline

1 Introduction

2 Relation Between Finite Element and Mimetic Finite Difference

3 Geometric Multigrid Methods

4 Conclusions and Future Work

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

10 / 25

SLIDE 29

Relation Between Finite Element and Mimetic Finite Difference curlh roth

Mimetic FDM and N´ ed´ elec FEM

N´ ed´ elec FEM Find uh ∈ VN

h , such that,

(rot uh, rot vh) + κ(uh, vh) = (f, vh), ∀ vh ∈ VN

h

where VN

h consists lowest order N´

ed´ elec finite elements.

8/ℎ2 4/ℎ2 4/ℎ2 −4/ℎ2 −4/ℎ2 8√3/3ℎ2 4√3/3ℎ2 4√3/3ℎ2 −4√3/3ℎ2 −4√3/3ℎ2

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

11 / 25

SLIDE 30

Relation Between Finite Element and Mimetic Finite Difference curlh roth

Mimetic FDM and N´ ed´ elec FEM

N´ ed´ elec FEM Find uh ∈ VN

h , such that,

(rot uh, rot vh) + κ(uh, vh) = (f, vh), ∀ vh ∈ VN

h

where VN

h consists lowest order N´

ed´ elec finite elements.

Equilateral triangle:

Mimetic FDM N´ ed´ elec FEM

8/ℎ2 4/ℎ2 4/ℎ2 −4/ℎ2 −4/ℎ2 8√3/3ℎ2 4√3/3ℎ2 4√3/3ℎ2 −4√3/3ℎ2 −4√3/3ℎ2

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

11 / 25

SLIDE 31

Relation Between Finite Element and Mimetic Finite Difference curlh roth

Mimetic FDM and Modified N´ ed´ elec FEM

General triangle:

𝑚𝑘𝑚

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

− 𝑚𝑗𝑚

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

𝑚𝑘𝑚

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

− 𝑚𝑗𝑚

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

2𝑚𝑗𝑘

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

𝛽 𝛾 𝑦𝑗

𝐸

𝑦𝑚

𝐸

𝑦𝑘

𝐸

1 𝑛𝑓𝑏𝑡(𝐸𝑙) − 1 𝑛𝑓𝑏𝑡(𝐸𝑙) 1 𝑛𝑓𝑏𝑡(𝐸𝑙) − 1 𝑛𝑓𝑏𝑡(𝐸𝑙) 2 𝑛𝑓𝑏𝑡(𝐸𝑙)

𝛽 𝛾

Mimetic FDM N´ ed´ elec FEM

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

12 / 25

SLIDE 32

Relation Between Finite Element and Mimetic Finite Difference curlh roth

Mimetic FDM and Modified N´ ed´ elec FEM

General triangle:

𝑚𝑘𝑚

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

− 𝑚𝑗𝑚

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

𝑚𝑘𝑚

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

− 𝑚𝑗𝑚

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

2𝑚𝑗𝑘

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

𝛽 𝛾 𝑦𝑗

𝐸

𝑦𝑚

𝐸

𝑦𝑘

𝐸

1 𝑛𝑓𝑏𝑡(𝐸𝑙) − 1 𝑛𝑓𝑏𝑡(𝐸𝑙) 1 𝑛𝑓𝑏𝑡(𝐸𝑙) − 1 𝑛𝑓𝑏𝑡(𝐸𝑙) 2 𝑛𝑓𝑏𝑡(𝐸𝑙)

𝛽 𝛾

Mimetic FDM N´ ed´ elec FEM

Consider a function u(x) ∈ VN

h ,

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

12 / 25

SLIDE 33

Relation Between Finite Element and Mimetic Finite Difference curlh roth

Mimetic FDM and Modified N´ ed´ elec FEM

General triangle:

𝑚𝑘𝑚

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

− 𝑚𝑗𝑚

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

𝑚𝑘𝑚

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

− 𝑚𝑗𝑚

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

2𝑚𝑗𝑘

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

𝛽 𝛾 𝑦𝑗

𝐸

𝑦𝑚

𝐸

𝑦𝑘

𝐸

1 𝑛𝑓𝑏𝑡(𝐸𝑙) − 1 𝑛𝑓𝑏𝑡(𝐸𝑙) 1 𝑛𝑓𝑏𝑡(𝐸𝑙) − 1 𝑛𝑓𝑏𝑡(𝐸𝑙) 2 𝑛𝑓𝑏𝑡(𝐸𝑙)

𝛽 𝛾

Mimetic FDM N´ ed´ elec FEM

Consider a function u(x) ∈ VN

h ,

u(x) =

(i,j)

DOF N

ij (u)ϕij =

(i,j)

xD

j

xD

i

u · eD

ij

ϕij
X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

12 / 25

SLIDE 34

Relation Between Finite Element and Mimetic Finite Difference curlh roth

Mimetic FDM and Modified N´ ed´ elec FEM

General triangle:

𝑚𝑘𝑚

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

− 𝑚𝑗𝑚

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

𝑚𝑘𝑚

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

− 𝑚𝑗𝑚

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

2𝑚𝑗𝑘

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

𝛽 𝛾 𝑦𝑗

𝐸

𝑦𝑚

𝐸

𝑦𝑘

𝐸

1 𝑛𝑓𝑏𝑡(𝐸𝑙) − 1 𝑛𝑓𝑏𝑡(𝐸𝑙) 1 𝑛𝑓𝑏𝑡(𝐸𝑙) − 1 𝑛𝑓𝑏𝑡(𝐸𝑙) 2 𝑛𝑓𝑏𝑡(𝐸𝑙)

𝛽 𝛾

Mimetic FDM N´ ed´ elec FEM

Consider a function u(x) ∈ VN

h ,

u(x) =

(i,j)

DOF N

ij (u)ϕij =

(i,j)

xD

j

xD

i

u · eD

ij

ϕij

=

(i,j)

(u · eD

ij )(xD ij ) lD ij

midpoint rule

ϕij

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

12 / 25

SLIDE 35

Relation Between Finite Element and Mimetic Finite Difference curlh roth

Mimetic FDM and Modified N´ ed´ elec FEM

General triangle:

𝑚𝑘𝑚

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

− 𝑚𝑗𝑚

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

𝑚𝑘𝑚

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

− 𝑚𝑗𝑚

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

2𝑚𝑗𝑘

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

𝛽 𝛾 𝑦𝑗

𝐸

𝑦𝑚

𝐸

𝑦𝑘

𝐸

1 𝑛𝑓𝑏𝑡(𝐸𝑙) − 1 𝑛𝑓𝑏𝑡(𝐸𝑙) 1 𝑛𝑓𝑏𝑡(𝐸𝑙) − 1 𝑛𝑓𝑏𝑡(𝐸𝑙) 2 𝑛𝑓𝑏𝑡(𝐸𝑙)

𝛽 𝛾

Mimetic FDM N´ ed´ elec FEM

Consider a function u(x) ∈ VN

h ,

u(x) =

(i,j)

DOF N

ij (u)ϕij =

(i,j)

xD

j

xD

i

u · eD

ij

ϕij

=

(i,j)

(u · eD

ij )(xD ij ) lD ij

midpoint rule

ϕij =

(i,j)

DOF MFD

ij

(u)lD

ij ϕij

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

12 / 25

SLIDE 36

Relation Between Finite Element and Mimetic Finite Difference curlh roth

Mimetic FDM and Modified N´ ed´ elec FEM

General triangle:

𝑚𝑘𝑚

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

− 𝑚𝑗𝑚

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

𝑚𝑘𝑚

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

− 𝑚𝑗𝑚

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

2𝑚𝑗𝑘

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

𝛽 𝛾 𝑦𝑗

𝐸

𝑦𝑚

𝐸

𝑦𝑘

𝐸

1 𝑛𝑓𝑏𝑡(𝐸𝑙) − 1 𝑛𝑓𝑏𝑡(𝐸𝑙) 1 𝑛𝑓𝑏𝑡(𝐸𝑙) − 1 𝑛𝑓𝑏𝑡(𝐸𝑙) 2 𝑛𝑓𝑏𝑡(𝐸𝑙)

𝛽 𝛾

Mimetic FDM N´ ed´ elec FEM

Consider a function u(x) ∈ VN

h ,

u(x) =

(i,j)

DOF N

ij (u)ϕij =

(i,j)

xD

j

xD

i

u · eD

ij

ϕij

=

(i,j)

(u · eD

ij )(xD ij ) lD ij

midpoint rule

ϕij =

(i,j)

DOF MFD

ij

(u)lD

ij ϕij

Modified basis functions: ϕmod

ij

= lD

ij ϕij

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

12 / 25

SLIDE 37

Relation Between Finite Element and Mimetic Finite Difference curlh roth

Mimetic FDM and Modified N´ ed´ elec FEM

General triangle:

𝑚𝑘𝑚

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

− 𝑚𝑗𝑚

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

𝑚𝑘𝑚

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

− 𝑚𝑗𝑚

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

2𝑚𝑗𝑘

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

𝛽 𝛾 𝑦𝑗

𝐸

𝑦𝑚

𝐸

𝑦𝑘

𝐸

𝑚𝑘𝑚

𝐸

𝑛𝑓𝑏𝑡(𝐸𝑙) − 𝑚𝑗𝑚

𝐸

𝑛𝑓𝑏𝑡(𝐸𝑙) 𝑚𝑘𝑚

𝐸

𝑛𝑓𝑏𝑡(𝐸𝑙) − 𝑚𝑗𝑚

𝐸

𝑛𝑓𝑏𝑡(𝐸𝑙) 2𝑚𝑗𝑘

𝐸

𝑛𝑓𝑏𝑡(𝐸𝑙)

𝛽 𝛾

Mimetic FDM Modified N´ ed´ elec FEM

Consider a function u(x) ∈ VN

h ,

u(x) =

(i,j)

DOF N

ij (u)ϕij =

(i,j)

xD

j

xD

i

u · eD

ij

ϕij

=

(i,j)

(u · eD

ij )(xD ij ) lD ij

midpoint rule

ϕij =

(i,j)

DOF MFD

ij

(u)lD

ij ϕij

Modified basis functions: ϕmod

ij

= lD

ij ϕij

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

12 / 25

SLIDE 38

Relation Between Finite Element and Mimetic Finite Difference curlh roth

Mimetic FDM and Modified N´ ed´ elec FEM

General triangle:

𝑚𝑘𝑚

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

− 𝑚𝑗𝑚

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

𝑚𝑘𝑚

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

− 𝑚𝑗𝑚

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

2𝑚𝑗𝑘

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

𝛽 𝛾 𝑦𝑗

𝐸

𝑦𝑚

𝐸

𝑦𝑘

𝐸

𝑚𝑘𝑚

𝐸

𝑛𝑓𝑏𝑡(𝐸𝑙) − 𝑚𝑗𝑚

𝐸

𝑛𝑓𝑏𝑡(𝐸𝑙) 𝑚𝑘𝑚

𝐸

𝑛𝑓𝑏𝑡(𝐸𝑙) − 𝑚𝑗𝑚

𝐸

𝑛𝑓𝑏𝑡(𝐸𝑙) 2𝑚𝑗𝑘

𝐸

𝑛𝑓𝑏𝑡(𝐸𝑙)

𝛽 𝛾

Mimetic FDM Modified N´ ed´ elec FEM

Consider a function u(x) ∈ VN

h ,

u(x) =

(i,j)

DOF N

ij (u)ϕij =

(i,j)

xD

j

xD

i

u · eD

ij

ϕij

=

(i,j)

(u · eD

ij )(xD ij ) lD ij

midpoint rule

ϕij =

(i,j)

DOF MFD

ij

(u)lD

ij ϕij

Modified basis functions: ϕmod

ij

= lD

ij ϕij

Modified test functions: ψmod

ij

= 1 lV

km

ϕij

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

12 / 25

SLIDE 39

Relation Between Finite Element and Mimetic Finite Difference curlh roth

Mimetic FDM and Modified N´ ed´ elec FEM

General triangle:

𝑚𝑘𝑚

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

− 𝑚𝑗𝑚

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

𝑚𝑘𝑚

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

− 𝑚𝑗𝑚

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

2𝑚𝑗𝑘

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

𝛽 𝛾 𝑦𝑗

𝐸

𝑦𝑚

𝐸

𝑦𝑘

𝐸

𝑚𝑘𝑚

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

− 𝑚𝑗𝑚

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

𝑚𝑘𝑚

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

− 𝑚𝑗𝑚

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

2𝑚𝑗𝑘

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

𝛽 𝛾 𝑦𝑗

𝐸

𝑦𝑚

𝐸

𝑦𝑘

𝐸

Mimetic FDM Modified N´ ed´ elec FEM

Consider a function u(x) ∈ VN

h ,

u(x) =

(i,j)

DOF N

ij (u)ϕij =

(i,j)

xD

j

xD

i

u · eD

ij

ϕij

=

(i,j)

(u · eD

ij )(xD ij ) lD ij

midpoint rule

ϕij =

(i,j)

DOF MFD

ij

(u)lD

ij ϕij

Modified basis functions: ϕmod

ij

= lD

ij ϕij

Modified test functions: ψmod

ij

= 1 lV

km

ϕij

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

12 / 25

SLIDE 40

Relation Between Finite Element and Mimetic Finite Difference curlh roth

Mimetic FDM and Modified N´ ed´ elec FEM

General triangle:

𝑚𝑘𝑚

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

− 𝑚𝑗𝑚

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

𝑚𝑘𝑚

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

− 𝑚𝑗𝑚

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

2𝑚𝑗𝑘

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

𝛽 𝛾 𝑦𝑗

𝐸

𝑦𝑚

𝐸

𝑦𝑘

𝐸

𝑚𝑘𝑚

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

− 𝑚𝑗𝑚

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

𝑚𝑘𝑚

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

− 𝑚𝑗𝑚

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

2𝑚𝑗𝑘

𝐸

𝑚𝑙𝑛

𝑊 𝑛𝑓𝑏𝑡(𝐸𝑙)

𝛽 𝛾 𝑦𝑗

𝐸

𝑦𝑚

𝐸

𝑦𝑘

𝐸

Mimetic FDM Modified N´ ed´ elec FEM

Consider a function u(x) ∈ VN

h ,

u(x) =

(i,j)

DOF N

ij (u)ϕij =

(i,j)

xD

j

xD

i

u · eD

ij

ϕij

=

(i,j)

(u · eD

ij )(xD ij ) lD ij

midpoint rule

ϕij =

(i,j)

DOF MFD

ij

(u)lD

ij ϕij

Modified basis functions: ϕmod

ij

= lD

ij ϕij

Modified test functions: ψmod

ij

= 1 lV

km

ϕij AMFD = D1 AN D2, where D1 = diag((lV

km)−1)

D2 = diag(lD

ij )

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

12 / 25

SLIDE 41

Relation Between Finite Element and Mimetic Finite Difference gradh divh

Raviart-Thomas FEM

4/3 2/3 2/3 2/3 2/3 2/3 2/3 2/3 2/3 2/3 2/3

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

13 / 25

SLIDE 42

Relation Between Finite Element and Mimetic Finite Difference gradh divh

Raviart-Thomas FEM

𝐼

𝑙 𝑛

RT basis functions on hexagons

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

13 / 25

SLIDE 43

Relation Between Finite Element and Mimetic Finite Difference gradh divh

Raviart-Thomas FEM

𝐼

𝑙 𝑛

RT basis functions on hexagons Find ϕkm ∈ V RT(H) (dim V RT(H) = 12) s.t.          ϕkm 2

∗→ min, ϕ ∗≃ ϕ L2, div ϕkm = c

ϕkm · njl = 0, ∀jl ∈ ∂H, jl = km
ϕkm · nkm = 1

(Ref: Kuznetsov, & Repin 2003; Boiarkine, Kuznetsov, & Svyatskiy 2007; Pasciak & Vassilevski, SISC, 2008)

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

13 / 25

SLIDE 44

Relation Between Finite Element and Mimetic Finite Difference gradh divh

Raviart-Thomas FEM

𝐼

𝑙 𝑛

RT basis functions on hexagons Find ϕkm ∈ V RT(H) (dim V RT(H) = 12) s.t.          ϕkm 2

∗→ min, ϕ ∗≃ ϕ L2, div ϕkm = c

ϕkm · njl = 0, ∀jl ∈ ∂H, jl = km
ϕkm · nkm = 1

What is c? c = divϕkm = 1 |H|

H

divϕkm = 1 |H|

∂H

ϕkm · njl

±1

= ± 1 |H|

(Ref: Kuznetsov, & Repin 2003; Boiarkine, Kuznetsov, & Svyatskiy 2007; Pasciak & Vassilevski, SISC, 2008)

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

13 / 25

SLIDE 45

Relation Between Finite Element and Mimetic Finite Difference gradh divh

Mimetic FDM and Modified Raviart-Thomas FEM

Mimetic FD:

2𝑚𝑙𝑛

𝑊

|𝐼|𝑚𝑗𝑘

𝐸

−𝑚𝑙𝑚

𝑊

|𝐼|𝑚𝑗𝑘

𝐸

−𝑚𝑙𝑚

𝑊

|𝐼|𝑚𝑗𝑘

𝐸

−𝑚𝑙𝑜

𝑊

|𝐼|𝑚𝑗𝑘

𝐸

−𝑚𝑙𝑜

𝑊

|𝐼|𝑚𝑗𝑘

𝐸

−𝑚𝑙𝑛

𝑊

|𝐼|𝑚𝑗𝑘

𝐸

−𝑚𝑙𝑛

𝑊

|𝐼|𝑚𝑗𝑘

𝐸

𝑚𝑙𝑚

𝑊

|𝐼|𝑚𝑗𝑘

𝐸

𝑚𝑙𝑜

𝑊

|𝐼|𝑚𝑗𝑘

𝐸

𝑚𝑙𝑚

𝑊

|𝐼|𝑚𝑗𝑘

𝐸

𝑚𝑙𝑜

𝑊

|𝐼|𝑚𝑗𝑘

𝐸

𝑦𝑗

𝐸

𝑦𝑘

𝐸

𝑦𝑜

𝑊

𝑦𝑚

𝑊

𝑦𝑛

𝑊

𝑦𝑙

𝑊

Raviart-Thomas FEM:

2 |𝐼| −1 |𝐼| −1 |𝐼| −1 |𝐼| −1 |𝐼| −1 |𝐼| −1 |𝐼| 1 |𝐼| 1 |𝐼| 1 |𝐼| 1 |𝐼|

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

14 / 25

SLIDE 46

Relation Between Finite Element and Mimetic Finite Difference gradh divh

Mimetic FDM and Modified Raviart-Thomas FEM

Mimetic FD:

2𝑚𝑙𝑛

𝑊

|𝐼|𝑚𝑗𝑘

𝐸

−𝑚𝑙𝑚

𝑊

|𝐼|𝑚𝑗𝑘

𝐸

−𝑚𝑙𝑚

𝑊

|𝐼|𝑚𝑗𝑘

𝐸

−𝑚𝑙𝑜

𝑊

|𝐼|𝑚𝑗𝑘

𝐸

−𝑚𝑙𝑜

𝑊

|𝐼|𝑚𝑗𝑘

𝐸

−𝑚𝑙𝑛

𝑊

|𝐼|𝑚𝑗𝑘

𝐸

−𝑚𝑙𝑛

𝑊

|𝐼|𝑚𝑗𝑘

𝐸

𝑚𝑙𝑚

𝑊

|𝐼|𝑚𝑗𝑘

𝐸

𝑚𝑙𝑜

𝑊

|𝐼|𝑚𝑗𝑘

𝐸

𝑚𝑙𝑚

𝑊

|𝐼|𝑚𝑗𝑘

𝐸

𝑚𝑙𝑜

𝑊

|𝐼|𝑚𝑗𝑘

𝐸

𝑦𝑗

𝐸

𝑦𝑘

𝐸

𝑦𝑜

𝑊

𝑦𝑚

𝑊

𝑦𝑛

𝑊

𝑦𝑙

𝑊

Raviart-Thomas FEM:

2 |𝐼| −1 |𝐼| −1 |𝐼| −1 |𝐼| −1 |𝐼| −1 |𝐼| −1 |𝐼| 1 |𝐼| 1 |𝐼| 1 |𝐼| 1 |𝐼|

Modified RT FEM:

New basis functions:

ϕmod

km

= lV

kmϕkm

New test functions:

ψmod

km

= 1 lD

ij

ϕkm AMFD = D1 ART D2 where D1 = diag((lD

ij )−1)

D2 = diag(lV

km)

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

14 / 25

SLIDE 47

Relation Between Finite Element and Mimetic Finite Difference Convergence

Error Analysis of Mimetic FDM based on FEM

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

15 / 25

SLIDE 48

Relation Between Finite Element and Mimetic Finite Difference Convergence

Error Analysis of Mimetic FDM based on FEM

curlh roth: ANUN = bN

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

15 / 25

SLIDE 49

Relation Between Finite Element and Mimetic Finite Difference Convergence

Error Analysis of Mimetic FDM based on FEM

curlh roth: ANUN = bN = ⇒ D1AND2

AMFD

D−1

2 UN UMFD

= D1bN

bMFD

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

15 / 25

SLIDE 50

Relation Between Finite Element and Mimetic Finite Difference Convergence

Error Analysis of Mimetic FDM based on FEM

curlh roth: ANUN = bN = ⇒ D1AND2

AMFD

D−1

2 UN UMFD

= D1bN

bMFD

= ⇒ UMFD = D−1

2 UN

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

15 / 25

SLIDE 51

Relation Between Finite Element and Mimetic Finite Difference Convergence

Error Analysis of Mimetic FDM based on FEM

curlh roth: ANUN = bN = ⇒ D1AND2

AMFD

D−1

2 UN UMFD

= D1bN

bMFD

= ⇒ UMFD = D−1

2 UN

Therefore, we have uMFD

h

(x) =

(i,j)

uMFD

ij

ϕmod

ij

(x) =

(i,j)

1 lD

ij

uN

ij lD ij ϕij =

(i,j)

uN

ij ϕij = uN(x)

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

15 / 25

SLIDE 52

Relation Between Finite Element and Mimetic Finite Difference Convergence

Error Analysis of Mimetic FDM based on FEM

curlh roth: ANUN = bN = ⇒ D1AND2

AMFD

D−1

2 UN UMFD

= D1bN

bMFD

= ⇒ UMFD = D−1

2 UN

Therefore, we have uMFD

h

(x) =

(i,j)

uMFD

ij

ϕmod

ij

(x) =

(i,j)

1 lD

ij

uN

ij lD ij ϕij =

(i,j)

uN

ij ϕij = uN(x)

Base on the standard error analysis for the N´ ed´ elec FEM, we automatically have u − uMFD

h

rot ≤ Ch

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

15 / 25

SLIDE 53

Relation Between Finite Element and Mimetic Finite Difference Convergence

Error Analysis of Mimetic FDM based on FEM

curlh roth: ANUN = bN = ⇒ D1AND2

AMFD

D−1

2 UN UMFD

= D1bN

bMFD

= ⇒ UMFD = D−1

2 UN

Therefore, we have uMFD

h

(x) =

(i,j)

uMFD

ij

ϕmod

ij

(x) =

(i,j)

1 lD

ij

uN

ij lD ij ϕij =

(i,j)

uN

ij ϕij = uN(x)

Base on the standard error analysis for the N´ ed´ elec FEM, we automatically have u − uMFD

h

rot ≤ Ch gradh divh: Based on the standard error analysis for the RT FEM, we automatically have u − uMFD

h

div ≤ Ch

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

15 / 25

SLIDE 54

Relation Between Finite Element and Mimetic Finite Difference Convergence

Error Analysis of Mimetic FDM based on FEM

curlh roth: ANUN = bN = ⇒ D1AND2

AMFD

D−1

2 UN UMFD

= D1bN

bMFD

= ⇒ UMFD = D−1

2 UN

Therefore, we have uMFD

h

(x) =

(i,j)

uMFD

ij

ϕmod

ij

(x) =

(i,j)

1 lD

ij

uN

ij lD ij ϕij =

(i,j)

uN

ij ϕij = uN(x)

Base on the standard error analysis for the N´ ed´ elec FEM, we automatically have u − uMFD

h

rot ≤ Ch gradh divh: Based on the standard error analysis for the RT FEM, we automatically have u − uMFD

h

div ≤ Ch Remarks:

Assume sufficiently smooth solution u(x) and regular domain Ω
Proper discretization for f(x) and mass lumping for the FEM (Brezzi, Fortin, &

Marini 2006)

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

15 / 25

SLIDE 55

Geometric Multigrid Methods

Outline

1 Introduction

2 Relation Between Finite Element and Mimetic Finite Difference

3 Geometric Multigrid Methods

4 Conclusions and Future Work

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

16 / 25

SLIDE 56

Geometric Multigrid Methods Geometric Multigrid

Multigrid for Mimetic FDM: curlh roth

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

17 / 25

SLIDE 57

Geometric Multigrid Methods Geometric Multigrid

Multigrid for Mimetic FDM: curlh roth

Approach: Use the relations between mimetic FDM and FEM

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

17 / 25

SLIDE 58

Geometric Multigrid Methods Geometric Multigrid

Multigrid for Mimetic FDM: curlh roth

Approach: Use the relations between mimetic FDM and FEM

Hierarchy of grids:

...

Components of the vector grid functions

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

17 / 25

SLIDE 59

Geometric Multigrid Methods Geometric Multigrid

Multigrid for Mimetic FDM: curlh roth

Approach: Use the relations between mimetic FDM and FEM

Hierarchy of grids:

...

Components of the vector grid functions

Choose components for GMG algorithm
Smoothers
Intergrid transfer operators: prolongation and restriction
Coarse grid problems
X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

17 / 25

SLIDE 60

Geometric Multigrid Methods Geometric Multigrid

Schwarz-type Smoother

Multiplicative/Additive Schwarz-type smoothers

Simultaneously update all the unknowns

around a vertex

Solve 6 × 6 systems of equations
Overlapping among the blocks

(Ref: Arnold, Falk, & Winther, Numer. Math. 2000)

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

18 / 25

SLIDE 61

Geometric Multigrid Methods Geometric Multigrid

Intergrid Transfer Operators

1 1 1/2 − sin(𝛽 + 𝛾) 2 sin 𝛽 1/2 − sin(𝛽 + 𝛾) 2 sin 𝛽 sin(𝛽 + 𝛾) 2 sin 𝛾 sin(𝛽 + 𝛾) 2 sin 𝛾 1/4 1/4 1/8 cos 𝛽 8 cos(𝛽 + 𝛾) 1/8 cos 𝛽 8 cos(𝛽 + 𝛾) − cos 𝛾 8 cos(𝛽 + 𝛾) − cos 𝛾 8 cos(𝛽 + 𝛾)

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

19 / 25

SLIDE 62

Geometric Multigrid Methods Geometric Multigrid

Intergrid Transfer Operators

Construct prolongation & restriction from the N´ ed´ elec canonical prolongation Q

1 1 1/2 − sin(𝛽 + 𝛾) 2 sin 𝛽 1/2 − sin(𝛽 + 𝛾) 2 sin 𝛽 sin(𝛽 + 𝛾) 2 sin 𝛾 sin(𝛽 + 𝛾) 2 sin 𝛾 1/4 1/4 1/8 cos 𝛽 8 cos(𝛽 + 𝛾) 1/8 cos 𝛽 8 cos(𝛽 + 𝛾) − cos 𝛾 8 cos(𝛽 + 𝛾) − cos 𝛾 8 cos(𝛽 + 𝛾)

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

19 / 25

SLIDE 63

Geometric Multigrid Methods Geometric Multigrid

Intergrid Transfer Operators

Construct prolongation & restriction from the N´ ed´ elec canonical prolongation Q

Prolongation: represent coarse grid basis as linear combination of fine

grid basis ϕH,mod

ij

= lD,H

ij

ϕH

ij

1 1 1/2 − sin(𝛽 + 𝛾) 2 sin 𝛽 1/2 − sin(𝛽 + 𝛾) 2 sin 𝛽 sin(𝛽 + 𝛾) 2 sin 𝛾 sin(𝛽 + 𝛾) 2 sin 𝛾 1/4 1/4 1/8 cos 𝛽 8 cos(𝛽 + 𝛾) 1/8 cos 𝛽 8 cos(𝛽 + 𝛾) − cos 𝛾 8 cos(𝛽 + 𝛾) − cos 𝛾 8 cos(𝛽 + 𝛾)

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

19 / 25

SLIDE 64

Geometric Multigrid Methods Geometric Multigrid

Intergrid Transfer Operators

Construct prolongation & restriction from the N´ ed´ elec canonical prolongation Q

Prolongation: represent coarse grid basis as linear combination of fine

grid basis ϕH,mod

ij

= lD,H

ij

ϕH

ij = lD,H ij

kl

qij

klϕh kl

1 1 1/2 − sin(𝛽 + 𝛾) 2 sin 𝛽 1/2 − sin(𝛽 + 𝛾) 2 sin 𝛽 sin(𝛽 + 𝛾) 2 sin 𝛾 sin(𝛽 + 𝛾) 2 sin 𝛾 1/4 1/4 1/8 cos 𝛽 8 cos(𝛽 + 𝛾) 1/8 cos 𝛽 8 cos(𝛽 + 𝛾) − cos 𝛾 8 cos(𝛽 + 𝛾) − cos 𝛾 8 cos(𝛽 + 𝛾)

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

19 / 25

SLIDE 65

Geometric Multigrid Methods Geometric Multigrid

Intergrid Transfer Operators

Construct prolongation & restriction from the N´ ed´ elec canonical prolongation Q

Prolongation: represent coarse grid basis as linear combination of fine

grid basis ϕH,mod

ij

= lD,H

ij

ϕH

ij = lD,H ij

kl

qij

klϕh kl =

kl

pij

kl

lD,H

ij

qij

kl

1 lD,h

kl ϕh,mod

kl

lD,h

kl

ϕh

kl =:

kl

pij

klϕh,mod kl

1 1 1/2 − sin(𝛽 + 𝛾) 2 sin 𝛽 1/2 − sin(𝛽 + 𝛾) 2 sin 𝛽 sin(𝛽 + 𝛾) 2 sin 𝛾 sin(𝛽 + 𝛾) 2 sin 𝛾 1/4 1/4 1/8 cos 𝛽 8 cos(𝛽 + 𝛾) 1/8 cos 𝛽 8 cos(𝛽 + 𝛾) − cos 𝛾 8 cos(𝛽 + 𝛾) − cos 𝛾 8 cos(𝛽 + 𝛾)

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

19 / 25

SLIDE 66

Geometric Multigrid Methods Geometric Multigrid

Intergrid Transfer Operators

Construct prolongation & restriction from the N´ ed´ elec canonical prolongation Q

Prolongation: represent coarse grid basis as linear combination of fine

grid basis ϕH,mod

ij

= lD,H

ij

ϕH

ij = lD,H ij

kl

qij

klϕh kl =

kl

pij

kl

lD,H

ij

qij

kl

1 lD,h

kl ϕh,mod

kl

lD,h

kl

ϕh

kl =:

kl

pij

klϕh,mod kl

Therefore, we have P = (D2,h)−1 Q (D2,H)

1 1 1/2 − sin(𝛽 + 𝛾) 2 sin 𝛽 1/2 − sin(𝛽 + 𝛾) 2 sin 𝛽 sin(𝛽 + 𝛾) 2 sin 𝛾 sin(𝛽 + 𝛾) 2 sin 𝛾 1/4 1/4 1/8 cos 𝛽 8 cos(𝛽 + 𝛾) 1/8 cos 𝛽 8 cos(𝛽 + 𝛾) − cos 𝛾 8 cos(𝛽 + 𝛾) − cos 𝛾 8 cos(𝛽 + 𝛾)

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

19 / 25

SLIDE 67

Geometric Multigrid Methods Geometric Multigrid

Intergrid Transfer Operators

Construct prolongation & restriction from the N´ ed´ elec canonical prolongation Q

Prolongation: represent coarse grid basis as linear combination of fine

grid basis ϕH,mod

ij

= lD,H

ij

ϕH

ij = lD,H ij

kl

qij

klϕh kl =

kl

pij

kl

lD,H

ij

qij

kl

1 lD,h

kl ϕh,mod

kl

lD,h

kl

ϕh

kl =:

kl

pij

klϕh,mod kl

Therefore, we have P = (D2,h)−1 Q (D2,H)

Restriction: similarly, R = (D1,H) QT (D1,h)−1

1 1 1/2 − sin(𝛽 + 𝛾) 2 sin 𝛽 1/2 − sin(𝛽 + 𝛾) 2 sin 𝛽 sin(𝛽 + 𝛾) 2 sin 𝛾 sin(𝛽 + 𝛾) 2 sin 𝛾 1/4 1/4 1/8 cos 𝛽 8 cos(𝛽 + 𝛾) 1/8 cos 𝛽 8 cos(𝛽 + 𝛾) − cos 𝛾 8 cos(𝛽 + 𝛾) − cos 𝛾 8 cos(𝛽 + 𝛾)

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

19 / 25

SLIDE 68

Geometric Multigrid Methods Geometric Multigrid

Intergrid Transfer Operators

Construct prolongation & restriction from the N´ ed´ elec canonical prolongation Q

Prolongation: represent coarse grid basis as linear combination of fine

grid basis ϕH,mod

ij

= lD,H

ij

ϕH

ij = lD,H ij

kl

qij

klϕh kl =

kl

pij

kl

lD,H

ij

qij

kl

1 lD,h

kl ϕh,mod

kl

lD,h

kl

ϕh

kl =:

kl

pij

klϕh,mod kl

Therefore, we have P = (D2,h)−1 Q (D2,H)

Restriction: similarly, R = (D1,H) QT (D1,h)−1

1 1 1/2 − sin(𝛽 + 𝛾) 2 sin 𝛽 1/2 − sin(𝛽 + 𝛾) 2 sin 𝛽 sin(𝛽 + 𝛾) 2 sin 𝛾 sin(𝛽 + 𝛾) 2 sin 𝛾 1/4 1/4 1/8 cos 𝛽 8 cos(𝛽 + 𝛾) 1/8 cos 𝛽 8 cos(𝛽 + 𝛾) − cos 𝛾 8 cos(𝛽 + 𝛾) − cos 𝛾 8 cos(𝛽 + 𝛾)

Prolongation Restriction

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

19 / 25

SLIDE 69

Geometric Multigrid Methods Geometric Multigrid

Coarse-grid Opertors

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

20 / 25

SLIDE 70

Geometric Multigrid Methods Geometric Multigrid

Coarse-grid Opertors

Rediscretization on the coarse grids

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

20 / 25

SLIDE 71

Geometric Multigrid Methods Geometric Multigrid

Coarse-grid Opertors

Rediscretization on the coarse grids satisfies AFD

H

= RAFD

h P

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

20 / 25

SLIDE 72

Geometric Multigrid Methods Geometric Multigrid

Coarse-grid Opertors

Rediscretization on the coarse grids satisfies AFD

H

= RAFD

h P

AFD

H

= D1,HAN

HD2,H

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

20 / 25

SLIDE 73

Geometric Multigrid Methods Geometric Multigrid

Coarse-grid Opertors

Rediscretization on the coarse grids satisfies AFD

H

= RAFD

h P

AFD

H

= D1,HAN

HD2,H = D1,HQTAN h QD2,H

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

20 / 25

SLIDE 74

Geometric Multigrid Methods Geometric Multigrid

Coarse-grid Opertors

Rediscretization on the coarse grids satisfies AFD

H

= RAFD

h P

AFD

H

= D1,HAN

HD2,H = D1,HQTAN h QD2,H

= D1,HQTD−1

1,h AMFD

D1,hAN

h D2,h D−1 2,hQD2,H

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

20 / 25

SLIDE 75

Geometric Multigrid Methods Geometric Multigrid

Coarse-grid Opertors

Rediscretization on the coarse grids satisfies AFD

H

= RAFD

h P

AFD

H

= D1,HAN

HD2,H = D1,HQTAN h QD2,H

= D1,HQTD−1

1,h AMFD

D1,hAN

h D2,h D−1 2,hQD2,H

=

R

(D1,HQTD−1

1,h) AMFD h P

(D−1

2,hQD2,H)

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

20 / 25

SLIDE 76

Geometric Multigrid Methods Geometric Multigrid

Coarse-grid Opertors

Rediscretization on the coarse grids satisfies AFD

H

= RAFD

h P

AFD

H

= D1,HAN

HD2,H = D1,HQTAN h QD2,H

= D1,HQTD−1

1,h AMFD

D1,hAN

h D2,h D−1 2,hQD2,H

=

R

(D1,HQTD−1

1,h) AMFD h P

(D−1

2,hQD2,H)

= RAMFD

h

P

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

20 / 25

SLIDE 77

Geometric Multigrid Methods Geometric Multigrid

Multigrid for Mimetic FDM: gradh divh

One difficulty: non-nested meshes

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

21 / 25

SLIDE 78

Geometric Multigrid Methods Geometric Multigrid

Multigrid for Mimetic FDM: gradh divh

One difficulty: non-nested meshes

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

One possible solution: go back to the triangle

!

! !

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

21 / 25

SLIDE 79

Geometric Multigrid Methods Geometric Multigrid

Multigrid for Mimetic FDM: gradh divh

One difficulty: non-nested meshes

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

One possible solution: go back to the triangle

!

! !

ϕmod

km

= lV

kmϕkm = lV km

i

αiϕRT

i

=

i

αilV

km ϕRT i

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

21 / 25

SLIDE 80

Geometric Multigrid Methods Geometric Multigrid

Multigrid for Mimetic FDM: gradh divh

One difficulty: non-nested meshes

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.2 0.4 0.6 0.8 1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

One possible solution: go back to the triangle

!

! !

ϕmod

km

= lV

kmϕkm = lV km

i

αiϕRT

i

=

i

αilV

km ϕRT i

Standard MG for H(div)
Auxiliary space preconditioner based on the regular decomposition of

H(div)

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

21 / 25

SLIDE 81

Geometric Multigrid Methods Geometric Multigrid

Numerical Experiments: curlh roth

W-cycle V-cycle ν ρ2g ρW

h

ρ3g ρV

h

1 0.331 0.330 0.337 0.334 2 0.124 0.124 0.133 0.132 3 0.070 0.069 0.072 0.071 4 0.045 0.045 0.052 0.052

Accurate predictions by Local

Fourier Analysis (LFA)

Optimal convergence of GMG
X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

22 / 25

SLIDE 82

Geometric Multigrid Methods Geometric Multigrid

Numerical Experiments: curlh roth

W-cycle V-cycle ν ρ2g ρW

h

ρ3g ρV

h

1 0.331 0.330 0.337 0.334 2 0.124 0.124 0.133 0.132 3 0.070 0.069 0.072 0.071 4 0.045 0.045 0.052 0.052

Accurate predictions by Local

Fourier Analysis (LFA)

Optimal convergence of GMG

Three-grid convergence rate predicted by LFA (different α and β)

Convergence factor deteriorates when

small angles appear

Possible to improve by using a relaxation

parameter ω (α = β = 80o: ρV

3g = 0.508, but

ρV

3g = 0.252 with ω = 1.35) 0.07 0.08 0.1 0.13 0.2 0.3 0.5 0.7 0.9

α β

5 15 25 35 45 55 65 75 85 5 15 25 35 45 55 65 75 85

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

22 / 25

SLIDE 83

Conclusions and Future Work

Outline

1 Introduction

2 Relation Between Finite Element and Mimetic Finite Difference

3 Geometric Multigrid Methods

4 Conclusions and Future Work

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

23 / 25

SLIDE 84

Conclusions and Future Work

Conclusions:

Relation between Mimetic FDM and Petrov-Galerkin FEM
Error Analysis for mimetic FDM can be derived from FEM framework
Efficient GMG for curlh roth and gradh divh can be designed with the help

from FEM

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

24 / 25

SLIDE 85

Conclusions and Future Work

Conclusions:

Relation between Mimetic FDM and Petrov-Galerkin FEM
Error Analysis for mimetic FDM can be derived from FEM framework
Efficient GMG for curlh roth and gradh divh can be designed with the help

from FEM Future Work:

Other finite element families on polytopal meshes

(Gillette, Rand, & Bajaj, 2014)

Applications in different physical models: Darcy’s flow, Maxwell’s

equation, etc

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

24 / 25

SLIDE 86

Conclusions and Future Work

Thank You!

Questions?

X. Hu

(Tufts) Multigrid for Mimetic FDM

Oct. 28, 2015

25 / 25